A NASA Perspective on Quantum Computing:Opportunities and Challenges

Rupak Biswas, Zhang Jiang, Kostya Kechezhi, Sergey Knysh, SalvatoreMandra, Bryan O’Gorman, Alejandro Perdomo-Ortiz, Andre Petukhov, JohnRealpe-Gomez, Eleanor Rieffel, Davide Venturelli, Fedir Vasko, Zhihui Wang

NASA Ames Research Center, Moffett Field, CA 94035

Abstract

In the last couple of decades, the world has seen several stunning instances

of quantum algorithms that provably outperform the best classical algorithms.

For most problems, however, it is currently unknown whether quantum algo-

rithms can provide an advantage, and if so by how much, or how to design

quantum algorithms that realize such advantages. Many of the most challeng-

ing computational problems arising in the practical world are tackled today by

heuristic algorithms that have not been mathematically proven to outperform

other approaches but have been shown to be effective empirically. While quan-

tum heuristic algorithms have been proposed, empirical testing becomes possible

only as quantum computation hardware is built. The next few years will be ex-

citing as empirical testing of quantum heuristic algorithms becomes more and

more feasible. While large-scale universal quantum computers are likely decades

away, special-purpose quantum computational hardware has begun to emerge

that will become more powerful over time, as well as some small-scale universal

quantum computers.

∗Rupak BiswasEmail address: first.last@nasa.gov (Rupak Biswas, Zhang Jiang, Kostya Kechezhi,

Sergey Knysh, Salvatore Mandra, Bryan O’Gorman, Alejandro Perdomo-Ortiz, AndrePetukhov, John Realpe-Gomez, Eleanor Rieffel, Davide Venturelli, Fedir Vasko, ZhihuiWang )

Preprint submitted to Elsevier April 18, 2017

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1. Introduction

In the last couple of decades, the world has seen several stunning instances

of quantum algorithms that provably outperform the best classical algorithms.

For most problems, however, it is currently unknown whether quantum algo-

rithms can provide an advantage, and if so by how much, or how to design

quantum algorithms that realize such advantages. Many of the most challeng-

ing computational problems arising in the practical world are tackled today by

heuristic algorithms that have not been mathematically proven to outperform

other approaches but have been shown to be effective empirically. While quan-

tum heuristic algorithms have been proposed, empirical testing becomes possible

only as quantum computation hardware is built. The next few years will be ex-

citing as empirical testing of quantum heuristic algorithms becomes more and

more feasible. While large-scale universal quantum computers are likely decades

away, special-purpose quantum computational hardware has begun to emerge

that will become more powerful over time, as well as some small-scale universal

quantum computers.

Successful NASA missions require solution of many challenging computa-

tional problems. The ambitiousness of such future missions depends on our

ability to solve yet more challenging computational problems to support better

and greater autonomy, space vehicle design, rover coordination, air traffic man-

agement, anomaly detection, large data analysis and data fusion, and advanced

mission planning and logistics. To support NASA’s substantial computational

needs, NASA Ames Research Center has a world-class supercomputing facility

with one of the world’s most powerful supercomputers. In 2012, NASA es-

tablished its Quantum Artificial Intelligence Laboratory (QuAIL) at Ames to

explore the potential of quantum computing for computational challenges aris-

ing in future agency missions. The following year, through a collaboration with

Google and USRA, NASA hosted one of the earliest quantum annealer proto-

types, a 509-qubit D-Wave II machine, which last summer was upgraded to a

1097-qubit D-Wave 2X system.

2

Because quantum annealers are the most advanced quantum computational

hardware to date, the main focus for the QuAIL team has been on both theoreti-

cal and empirical investigations of quantum annealing, from deeper understand-

ing of the computational role of certain quantum effects to empirical analyses of

quantum annealer performance on small problems from the domains of planning

and scheduling, fault diagnosis, and machine learning. This paper will concen-

trate on the team’s quantum annealing work, with only brief mention of research

related to capabilities of other near-term quantum computational hardware that

will be able to run quantum heuristic algorithms beyond quantum annealing.

For information on quantum computing more generally, and other algorithms,

both heuristic and non, see quantum computing texts such as [1].

The power of quantum computation comes from encoding information in

a non-classical way, in qubits, that enable computations to take advantage of

purely quantum effects, such as quantum tunneling, quantum interference, and

quantum entanglement, that are not available classically. The beauty of quan-

tum annealers is that users can program them without needing to know about

the underlying quantum mechanical effects. Knowledge of quantum mechanics

aids in more effective programming, just as an understanding of compilation

procedures can aid classical programming, but it is not necessary for a basic

understanding.

For this reason, the first three sections consist of an overview of quantum

annealing (Sec. 2), a description of how to program a quantum annealer (Sec. 3),

and a high-level review of our exploration of three potential application areas

for quantum annealing (Sec. 4). The quantum effects involved are only lightly

mentioned, so these sections should be easily accessible to computer scientists

without any knowledge of quantum mechanics or quantum computing. Sec. 5,

which examines the role various physical processes play in quantum anneal-

ing, requires more physics knowledge for a full understanding, as does Sec. 6

that discusses hardware, though a classically-trained computer scientist with-

out knowledge of quantum mechanics can get a high-level understanding. We

conclude with a brief section summarizing the outlook for the future.

3

2. Quantum annealing

Quantum annealing [2, 3] is a metaheuristic optimization algorithm that

makes use of quantum effects such as quantum tunneling and interference. It

is one of the most accessible quantum algorithms to people versed in classical

computing because of its close ties to classical optimization algorithms such

as simulated annealing and because the most basic aspects of the algorithm

can be captured by a classical cost function and parameter setting. Quantum

annealers are special-purpose quantum computational devices that can run only

the quantum annealing metaheuristic. For readers not familiar with quantum

annealing in physics, we refer to Sec. 5.1 for a general introduction.

Quantum annealers are designed to minimize Quadratic Unconstrained Bi-

nary Optimization (QUBO) problems; i.e., the cost function is of the form

C(x) =∑i

aixi +∑i<j

bi,jxixj , (1)

where {ai, bi,j} are real coefficients and x ∈ {0, 1}n is a vector of binary-valued

variables. An application problem must be mapped to a QUBO before it can

be solved on a quantum annealer. For application problems with constraints,

the cost function is supplemented with penalty terms that penalize bit strings

that do not correspond to valid solutions.

The simplicity of the QUBO formalism belies its expressivity. There exist

many techniques for mapping more complicated problems to QUBO:

• A wide class of optimization problems of practical interest can be ex-

pressed in terms of cost functions that are polynomials over finite sets of

binary variables. Any such function can be re-expressed, through degree-

reduction techniques using ancilla variables, as quadratic functions over

binary variables. We describe such degree-reduction technique in our sec-

tion on the CNF mapping of planning problems to QUBO.

• Cost functions involving non-binary, but finite-valued, variables can be

rewritten in terms of binary variables alone, and optimization problems

4

with constraints can often be written entirely in terms of cost functions

over binary variables through the introduction of slack variables.

For these reasons, the QUBO setting is more general than it may seem. We

give examples of QUBO mappings for different applications domains in later

sections.

Current quantum annealers such as the D-Wave 2X are fabricated using su-

perconducting materials and operated at tens of milli-Kelvin temperatures. The

processors make use of superconducting flux qubits [4] that are superconductor

loops sandwiched with Josephson junctions, engineered so that when an exter-

nal flux is applied, a persistent current appears in the loop. The computational

basis of the qubit is the clockwise and counter-clockwise flow of the currents,

corresponding to values of +1 and -1, respectively, of the spin variable sj for

qubit j.

An Ising Hamiltonian

H1 =∑j

hjsj +∑i,j

Ji,jsisj (2)

can be programmed on the D-Wave system by setting the values of the flux biases

hj on each qubit sj and couplings Ji,j between qubits. A mapping sj = 2xj − 1

relates an Ising Hamiltonian to a QUBO form. Because only select couplers

are implemented in the hardware, only certain quadratic terms can be directly

implemented. Embedding, using multiple qubits to represent a single binary

variable, is necessary to implement arbitrary QUBOs, a topic we will return to

when we discuss programming quantum annealers in more depth.

Quantum annealing is carried out by evolving the system under the time-

dependent Hamiltonian

H(t) = A(s)H0 +B(s)H1 (3)

where H1 is the problem Hamiltonian in QUBO form and H0 is the initial

Hamiltonian, which in current annealers is fixed and cannot be set by the pro-

grammer. Generally, the Hamiltonian H0 is chosen to have a simple energy

5

Figure 1: Typical annealing profile A(s) and B(s).

landscape so that an unsophisticated relaxation process will efficiently put the

system in low energy states. During the anneal, H0 is gradually changed until

it becomes H1. The intuition is that if the system starts in low energy states

and the change is smooth enough, the system will end up in low energy states

of the final Hamiltonian, just as a top spinning on a tray will continue to spin

when the tray is moved as long as the change in position is smooth enough. The

functions A(s) and B(s) are generally chosen in a way that H0 dominates at

s = 0 and H1 dominates at s = 1 (see Fig. 1). Current annealers provide a range

of total anneal times tf , where s = t/tf , enabling traversals at different speeds.

On the D-Wave 2X housed at NASA, the annealing time can be chosen in a

range from 5 µs to 2 ms. Future annealers may allow programmers to choose

A(s) and B(s), but they are currently fixed in the D-Wave 2X.

When viewed as an algorithm for exploring the landscape defined by the cost

function to find a global minimum, quantum annealing resembles a commonly

used classical algorithm for optimization: simulated annealing. While in simu-

lated annealing thermal fluctuation provides the mobility over energy barriers

6

between local minima, quantum annealing has an additional source of mobility:

quantum fluctuations that facilitate tunneling through the barriers. Such quan-

tum fluctuations are realized through H0 which serves as a driver Hamiltonian

responsible for quantum fluctuations because it does not commute with the tar-

get Hamiltonian H1. As the anneal continues, the driver term is reduced, slowly

turning off the fluctuations, as the problem Hamiltonian’s strength increases.

Quantum annealing should not be confused with adiabatic quantum compu-

tation which is known to support universal quantum computing. The problem

Hamiltonian in quantum annealing typically is a classical Hamiltonian. While

adiabatic quantum computation also interpolates between an initial and final

Hamiltonian, the final Hamiltonian can be highly non-classical with no analo-

gous classical cost function, thus enabling much more general sorts of quantum

computations.

3. Programming a quantum annealer

This section discusses the two main steps in programming a quantum an-

nealer: mapping the problems to QUBO; and embedding , which takes these

hardware-independent QUBOs to other QUBOs that match the specific quan-

tum annealing hardware that will be used.

3.1. Mapping

For a cost function not natively in QUBO form, the typical procedure to

map the problem into QUBO is to properly choose binary variables, formulate

constraints, and embed the violation of constraints as energy penalties. We

illustrate this process with an example from Ref. [5].

Example: In a graph coloring problem, the task is to determine whether

each vertex of a graph G(V,E) can be colored from a set C so that no two

vertices connected by an edge have the same color. The goal is to formulate

a cost function such that the minimum is 0. One way to choose the binary

variable is to use xv,c = 0 or 1 to express whether vertex v is assigned color

7

c. The ensuing constraints would be: (1) Each vertex needs to be assigned

exactly one color that can be expressed in binary form as (∑c xv,c − 1)2. (2)

Connected vertices cannot share the same color; otherwise, the energy penalty

is raised,∑c

∑v,v′∈E xv,cxv′,c. The cost function expressed in QUBO is then

H =∑v(∑c xv,c−1)2+

∑c

∑v,v′∈E xv,cxv′,c. When no requirement is violated,

the cost function has value 0, which is the ground state of H.

In this example, the cost function H is naturally quadratic. More gener-

ally, the cost functions of many optimization problems can be expressed as

higher-degree polynomials of the binary variables (PUBOs). Degree-reduction

techniques can then be applied to recast a PUBO as QUBO, usually at the price

of adding ancilla variables [6].

3.2. Embedding

Because the physical hardware has limited connectivity, there usually does

not exist a direct one-to-one mapping between the QUBO binary variables and

the physical qubits so that each binary term in the QUBO corresponds to a

pair of connected qubits. To obtain the needed connectivity in the embeddable

QUBO, an additional step is required. Unlike the mapping step, the embedding

step is hardware dependent. A cluster of qubits {yi,k} connected to each other

in the hardware graph will represent a single variable xi. For any term xixj in

the mapped QUBO, there is a connection in the embeddable QUBO between

one of the qubits in the cluster for xi and one qubit in the cluster for xj . Minor

embedding is the process of determining a cluster for each binary variable in

the problem QUBO [7]. The problem of finding the optimal minor embedding

is itself NP-complete, but fortunately it is not necessary to find the optimal

embedding. In general, for planar architectures, there are straightforward, fast

algorithms to embed an N -variable problem in hardware consisting of no more

than N2 physical qubits [7, 8, 9]. In the near term, while the hardware is so

qubit constrained, heuristic algorithms [10] are used to try to minimize resources

and maximize the size of the problems embeddable on the machine.

To encourage the qubits in the cluster to all take the same value by the end

8

of the anneal so that the value of the variable they represent is unambiguous, the

embeddable QUBO also includes constraint terms JF yi,pyi,q for any pair p, q of

qubits in the cluster that are connected to each other, where JF is the strength of

the coupling. This is to ensure that in the most energy-favorable configuration,

all qubits in the cluster take the same value. The Hamiltonian obtained from

the embeddable QUBO shares the same ground state energy as the Hamiltonian

from the mapped QUBO, but conforms to the hardware architecture. The

higher energy spectrum may be considerably altered, so different embeddings

can significantly affect performance.

The optimal strength of JF is a subject of extensive research [5, 11, 12]. One

might think it should be as high as possible to force the qubits to all take the

same value at the end, but in practice there is a sweet spot. Coupling strengths

that are too high degrade performance. Intuitively, a high coupling strength

makes it harder to change the value of a variable in the cluster once they take

on a value that is not, ultimately, optimal, though the actual quantum dynamics

are more complicated than this simple explanation.

The layout of the qubits and couplers of a D-Wave quantum annealer is a

n × n lattice of unit cells called a Chimera graph. Each unit cell is composed

of a bi-partite graph of 8 qubits. A schematic diagram of the graph formed by

9 cells is shown in Fig. 2. The current D-Wave machine at NASA has 12 × 12

such units and a total of 1152 qubits, of which 1097 are working. Each qubit

is coupled to at most 6 other qubits, 4 within its own unit cell and 2 to qubits

in its neighboring cells. To embed a generic QUBO of N variables, N2 qubits

and couplers are needed in the worst case so that each binary variable can

be represented by N physical qubits and effectively couple to all other binary

variables. As an illustration, Fig. 3 shows an example of embedding a triangle

onto a bi-partite graph.

When an Ising problem is programmed to the chip, errors due to noise or

manufacturing miscalibration associated with the bias fields (h’s) and couplers

(J ’s) would affect the annealing performance. Simple offset errors can be cor-

rected through software, but more complicated errors are harder to mitigate.

9

Figure 2: Nine unit cells in a Chimera graph.

One strategy is to repeat the annealing with a gauge-transformed Hamiltonian

in which the states used to represent 0 and 1 are swapped. The qubits are

encoded into s′j = gjsj where gj = ±1, and the biases and couplers are ac-

cordingly set as h′j = gjhj and J ′i,j = gigjJi,j . The resulting Hamiltonian

H ′ =∑j h′js′j +∑i,j J

′i,js′is′j , which is equal to the original Hamiltonian, is sent

to the annealer and the solution obtained is then decoded using sj = gjs′j . One

set of parameters {gj} is called a gauge. In the absence of errors, the annealing

results for H and H ′ should be the same while the actual performance could be

gauge-dependent. Success probabilities averaged over a set of gauges are typi-

cally used. Various error suppression and correction strategies exist, both fully

quantum [13], a mix of quantum and classical [14], and a more recent quan-

tum approach [15]. Once the problem is programmed, the annealing is repeated

multiple times (typically thousands to millions), and each time the final state

measured in the computational basis is recorded.

10

a b

c d

1

23

Figure 3: Schematics of embedding the Hamiltonian H = J1,2s1s2 + J1,3s1s3 + J2,3s2s3

on a graph. Left: Triangle graph to be embedded. Right: Graph after embedding on a bi-

partite graph of size 4. The variable s1 is represented by two physical qubits sa and sb with

a strong ferro-magnetic coupling JF < 0. The Hamiltonian after embedding is Hembed =

JF sasb + J1,2sasd + J1,3sbsc + J2,3scsd.

4. Applications

In this section, we give a high-level overview of our in-depth studies of three

potential applications areas: planning and scheduling, fault diagnosis, and ma-

chine learning. Further technical details can be found in the publications refer-

enced in each section.

4.1. Quantum annealing for planning and scheduling

Automated planning and scheduling has many applications, from logistics,

air traffic control, and industrial automation to conventional military missions,

resource allocation, and assistance in disaster recovery. Many of the challenges

in autonomous operations include significant planning and multi-agent coordi-

nation tasks in which operational teams must generate courses of action prior

to the event and adjust those plans as new information becomes available or

unexpected events occur.

Many planning and scheduling problems are very challenging to solve; as the

number of events to plan or schedule grows, the number of possible solutions

grows exponentially. These problems are often NP-hard or harder, and are

currently tackled by classical heuristic algorithms. The emergence of quantum

annealing hardware allows the exploration of quantum heuristic approaches to

these problems [3], with the objectives to search for significant improvements

11

over existing techniques in the efficiency with which good plans can be found, or

in finding better plans that satisfy more constraints, and/or in greater diversity

in the plans found.

Given the severe limitation in quantum memory of current quantum anneal-

ers, in order to benchmark the machines, it is imperative to find prescriptions to

identify small problems that exhibit signature of hardness. Currently, the most

common approach to designing benchmark planning problems is to extract solv-

able problems from real-world applications. This approach has the benefit of

tuning algorithms toward the applications from which the benchmark problems

are obtained. A complementary approach is to design parametrized families

that capture aspects of practical planning problems and can be shown to be

intrinsically hard. Such families support focused examination of these aspects,

small problems that can be meaningfully considered to be hard, and scaling

analyses with respect to size. Families of small but hard problems are criti-

cal for present research into quantum annealing because the current quantum

annealers can handle only small problems. Families we have designed for the

purpose of assessing the performance of quantum annealers have proved useful

in distinguishing the strengths and weaknesses of state-of-the-art planners [16].

4.1.1. QUBO formulation of general planning problems

Classical planning problems are expressed in terms of binary state variables

and actions. Examples of state variables in the domain of autonomous rover

navigation are “Rover R is in location X” and “Rover R has a soil sample from

location X,” which may be True or False. Actions consist of two lists, a set of

preconditions and a set of effects (see Fig. 4). The effects of an action consists

of a subset of state variables with the values they take on if the action is carried

out. For example, the action “Rover R moves from location X to location Y ”

has one precondition, “Rover R is in location X = True” and has two effects

“Rover R is in location X = False” and “Rover R is in location Y = True.”

A specific planning problem specifies an initial state, with values specified

for all state variables, and a goal , specified values for one or more state variables.

12

Figure 4: (a) Pictorial view of a planning problem. The initial state (e.g., Rover behind

the rocks without sample) is specified by assigning True (1) or False (0) to state variables

(named A-J in this simplified example). The planning software navigates a tree, where a

path represents a sequence (with possible repetitions) of actions selected from a pool (colors).

Each action has preconditions on the state variables (e.g., moves can be done around the rocks

but not through them) that need to be satisfied for the actions to be executed (the circles

under the state variables in the search tree need to be True) and has an effect on the state

(colored variables in shaded regions of the new state have changed values). A valid search plan

(multiple valid plans are possible) will reach the goal state (e.g., Rover in front of the rocks

with a sample collected). (b) Direct time-indexed QUBO structure for a planning problem

with only positive preconditions and goals. Each node represents a state variable (left) or an

action (right) at any given time t. Time flows from top to bottom, and variables y(t)i for the

actions at time t are shown between the state variables x(t−1)i for one time step and the state

variables x(t)i for the next time step. The node grayscale intensity represents the magnitude

of local field (bias) hi applied to a given qubit i, and the double contour in a node indicates

a negative bias.

As for preconditions, goals are conventionally positive, so the specified value for

the goal variables is True. Generally, the goal specifies values for only a small

subset of the state variables. A plan is a sequence of actions. A valid plan, or a

solution to the planning problem, is a sequence of actions A1, ..., AL such that

the state at time step ti−1 meets the preconditions for action Ai, the effects of

action Ai are reflected in the state at time step ti, and the state at the end has

all of the goal variables set to True.

Ref. [5] discusses a general QUBO formulation of planning problems (see

13

Fig. 4(b)). If the original planning problem has N state variables and we are

looking for a plan of length L, then the QUBO problem will have N(L + 1)

binary variables x(t)i , where t ∈ {0, . . . , L} is the time index, and i is the index

of the state variable in the original planning problem. In addition, if the original

planning problem has M possible actions, we will have LM additional binary

variables y(t)j which indicate whether the jth action is carried out at time step

t or not. A QUBO can then be defined in terms of these variables, with terms

capturing the goal, precondition, effect, single-action, and no-op (no variable

change without an action) constraints:

H = H ′goal +Hno− op +H ′precond +Heffects +Hsingle-action. (4)

Ref. [5] describes a somewhat more general cost function that supports multiple

actions per time step.

4.1.2. Advanced scheduling applications

Scheduling was recognized early on as one the most promising near-term tar-

gets for quantum annealing due to its efficient quadratic time-indexed Mixed-

Integer Linear Programming formulation. Furthermore, there is a rich litera-

ture of complex pre-processing and hybrid classical techniques. Using this direct

quadratic formulation of scheduling instead of the most general planning formu-

lation leads to very significant performance advantages in runs of the D-Wave

machines [5].

Scheduling formalizes problems dealing with the optimal allocation of re-

sources (machines, people) to tasks (jobs) over time, under various constraints

and figures of merit. In one direct QUBO formulation, a bit is associated to

the execution of a given job in a given machine (out of M possible) at a given

time (discretized in T slots), allowing for very efficient mappings on current

quantum annealers supporting two-body Ising-type interactions, using NMT

qubits, where N is the number of jobs. While objective functions of the pri-

ority maximization type are easily implementable as linear penalty functions

requiring only local fields on the corresponding logical bits, objectives requiring

14

makespan minimization require a more involved encoding with either T ancilla

clock variables highly connected to the qubits relative to the jobs scheduled last,

or by complementing the quantum solver with guidance from classical methods,

such as binary search [12].

Many planning and scheduling problems are of such scale and complexity

that they are by necessity solved in pieces, and so quantum hardware can be

naturally integrated into the solution of such problems. Hybrid solvers employ-

ing quantum annealing together with classical methods are particularly suited

to scheduling applications, because the state-of-the-art approaches for specific

scheduling problems are typically combining different approaches in a mod-

ular way, and decompositions can be employed to get around programming

bottlenecks such as high connectivity, precision requirements, continuous con-

straints, or to employ quantum annealing as a heuristic module of a complete

solver [17, 18]. As a heuristic module of a complete solver, quantum anneal-

ing enables more directed search of the solution space. Building a complete

solver out of a probabilistic quantum subroutine requires non-trivial classical

co-processing, but recent work has shown that it can be done successfully. In

particular, partial solutions returned by a quantum solver can be used to derive

bounds on the optimum value of the function to be optimized, and therefore

focus on the most promising or neglect the least promising parts of the solution

space.

Recent work on the application of quantum annealing to scheduling includes

programming and benchmarking quantum annealers on small problems from

the domains of graph coloring [5], job shop scheduling [12], Mars lander activity

scheduling [17], air traffic runway landing [18], and alternative resource schedul-

ing [18]. The question of speedup with respect to purely classical methods

are inconclusive due to the small size of the problems implementable on cur-

rent quantum annealers and the inefficiency of embedding techniques [5]. This

body of work has identified precision and connectivity requirements that sug-

gest future generations of annealers may be able to solve currently intractable

scheduling problems within a decade.

15

Planned technological advances in quantum annealing architectures will also

make possible tighter integration of quantum and classical components in the

hybrid approaches discussed above, both through more programmable devices

that allow for greater flexibility as subroutines and through application-specific

devices that maximize the effectiveness of particular algorithms. In future, we

expect quantum hardware to be integrated into larger systems much as graphical

processing units are today [19].

4.2. Fault detection and diagnostics of graph-based systems

Another application domain we have studied with quantum annealing de-

vices is the diagnostics of electrical power-distribution systems (EPS); a collab-

oration between QuAIL and the Discovery and System Health (DaSH) technical

area at NASA Ames. Diagnosing the minimal number of faults capable of ex-

plaining a set of given observations, e.g., from sensor readouts, is a hard combi-

natorial optimization problem usually addressed with artificial intelligence tech-

niques. In [20], we presented the first application of the Combinatorial Problem

→ QUBO Mapping → Direct Embedding process where we were able to em-

bed instances with sizes comparable to those found in real-world problems. We

demonstrated problem instances with over 100 electrical components (includ-

ing circuit breakers and sensors) running on a quantum annealing device with

509 quantum bits. In comparison, the number of components in the electrical

circuits used for diagnostics competitions from NASA’s Advanced Diagnostics

and Prognostics Testbed (ADAPT) ranges between 40 and 100 [21].

4.2.1. QUBO formulation

As shown in Fig. 5(a), there are two types of components. The first are

circuit breakers (CB), which in their healthy mode allow the flow of current, and

are illustrated as the nodes of the quaternary tree. We denote them by the set

of binary variables {xi}, with xi = 1 (xi = 0) corresponding to CB i in a healthy

(faulty) state. The other component type is the sensor or ammeter, which is not

only another electrical component that could potentially malfunction, but also

16

forms part of the observations from which one is asked to perform the diagnosis

of the electrical network. Therefore, for each ammeter, we have an observation

parameter and a status variable indicating its healthy or faulty state. The

observations (or readouts) are part of the problem definition and provided as

input parameters. We denote this set of binary parameters {li}, with li = 1

(li = 0) if the i-th ammeter is showing a High (Low) readout. Similar to the {xi}variables for the CBs, the uncertainty in the ammeter readouts is introduced

by assigning to them a set of binary variables, {yi}, with yi = 1 (yi = 0)

corresponding to ammeter i in a healthy (faulty) state.

The goal is to find the minimum number of faults in the electrical compo-

nents, either in the CBs and/or the ammeters, consistent with the circuit layout

and the readouts. We solve this as a minimization problem over the pseudo-

Boolean function Hproblem({xi}, {yi}; {li}), whose construction is explained be-

low. After Hproblem is transformed into its QUBO form, we can subsequently

use the quantum annealer to find the assignment for each of the {xi} and {yi}.The construction of the pseudo-Boolean function contains two contributions:

Hproblem = HnumFaults +Hconsist. (5)

Hconsist is constructed such that it is 0 whenever the prediction from the as-

signment of all the {xi} and {yi} is consistent with the readouts {li} from the

ammeters, and greater than 0 when the readouts and the prediction, given the

{xi} and {yi} assignments, do not match. Consider the set Pi as the set of CB

indices in the path from the root node (CB 1) where power is input, all the way

to the CB connected to the i-th ammeter. Thus, for the network in Fig. 5(a),

P1 = {1, 2, 6}, P2 = {1, 2, 7}, · · · , and P16 = {1, 5, 21}. If we denote the number

of paths as npaths (equals the number of ammeters in this network), one can

construct Hconsist as:

Hconsist = λpath

npaths∑i=1

yigi, fi({xj}j∈Pi) =

∏j∈Pi

xj , (6)

with gi = li + fi − 2fili, a binary function with gi = 0 when the prediction fi,

17

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46

(b)$QUBO$form$

(a)$Computa2onal$problem$ (c)$Hardware$embedding$

46#logical#qubits#

81#hardware#qubits#

Sensors:##

CBs:#

0# 1#0# 1# 0# 0# 1# 1# 1# 1#1# 1# 1# 1#0# 0#Obs.:#

Figure 5: General scheme of an experimental setup for the diagnosis of multiple faults with

a quantum annealer. (a) A possible realization of the diagnosis of multiple faults in an EPS

network with one power source, 21 CBs and 16 sensors or ammeters. The orange crosses

indicate faulty electrical components (xi = 0). In this particular instance of 6 faults, a

plausible explanation of the readouts places one of the faults on a CB and the remaining 5 on

the ammeters. However, this is only one of the 26 six-fault explanations that are equally likely

in this case. (b) QUBO form of the problem where coupling between two logical qubits is

represented as edges. (c) The subsequent embedding into the Chimera graph usually requires

more variables since some logical qubits are represented by several physical qubits (depicted

here as nodes in the graph) due to the sparse connectivity of the hardware graph. In this

problem, 81 physical qubits are needed to implement the QUBO with 46 logical variables.

18

based only on the CB statuses in the path Pi, is consistent with the readouts li,

and gi = 1 when the prediction and the readout are in disagreement. In other

words, gi = xor(fi, li).

HnumFaults is proportional to the number of faults (whenever xi = 0 or

yi = 0) in the electrical network:

HnumFaults = λCBfaults

nCB∑i=1

(1− xi) + λsensorfaults

nsensor∑i=1

(1− yi), (7)

and when combined with Hconsist, as written in Eq. (5), defines the problem en-

ergy function to be minimized by favoring the minimal set of faulty components

that are simultaneously consistent with the observations in the outermost sen-

sors. A thorough discussion on setting the values of all the penalties is provided

in [20].

Notice the pseudo-Boolean Hconsist is a high-degree polynomial, and for this

particular network, the order of the polynomial is related to the depth of the

tree. We can reduce the degree of the polynomial to a quadratic expression,

HQUBO, with the overhead of adding more binary variables, while conserving

the global minimum of the original function, H({xi}, {yi}; {li}). Further details

on the techniques used for this reduction are provided in [20, 22].

Assuming it requires nA ancilla variables {ai} to reduce the high-degree

polynomial to the quadratic expression, we can relabel the CB, sensor, and

ancilla variables, {xi}, {yi}, and {ai}, respectively, into a new set of binary

variables {qi} for i = 1, 2, · · · , nl, with nl = nCB + nsensor + nA as the total

number of logical qubits. The final quadratic cost function to be minimized can

then be written as

HQUBO({qi}) = E0QUBO +

∑i,j

Qi,jqiqj

= E0QUBO + qT ·Q · q.

(8)

As shown in Fig. 5, this expression can be represented as a graph with the

number of vertices equal to the number of logical qubits nl corresponding to the

set of variables {qi}. In this representation, Qi,i can be treated as the weights on

19

the vertices, while Qi,j are the weights for the edges representing the couplings

between variables i and j (see Fig. 5). Notice that since q2i = qi, the expression

qT ·Q ·q contains both linear terms Qi,i, and quadratic terms, Qi,j , when i 6= j.

E0QUBO corresponds to the constant independent term.

Although the problems studied in [20] are simpler than typical real-world

instances, we believe that they still capture some non-trivial features, such as

the inclusion of uncertainty in the sensor readouts. Of course, aiming to embed

all the details from realistic scenarios will require significantly more qubits and

also depend on the specific network/problem to be solved.

As another realization of the fault detection application, the QuAIL team

is examining combinational digital circuits [23], a more realistic scenario used

to benchmark codes devoted to solving diagnostics related problems [21]. Pre-

liminary results look very promising and harder than any other benchmarks

reported in the literature and used to address the question of quantum speedup

in quantum annealers.

4.3. Sampling and machine learning applications

Sampling from high-dimensional probability distributions is at the core of a

wide spectrum of computational techniques with important applications across

science, engineering, and society. Examples include deep learning, probabilistic

programming, and other machine learning and artificial intelligence applications.

Much of the record-breaking performance of classical machine learning algo-

rithms regularly reported in the literature pertains to task-specific supervised

learning algorithms [24]. Unsupervised learning algorithms are more human-

like, and in principle more general and powerful, but their development has

been lagging due to the intractability of traditional sampling techniques such

as Markov Chain Monte Carlo (MCMC). Indeed, as leading researchers in the

field have pointed out [24], future success of unsupervised learning algorithms

requires breakthroughs in efficient sampling algorithms. Quantum annealing

holds the potential to sample more efficiently and from more complex prob-

abilistic models, which would significantly advance the field of unsupervised

20

learning.

4.3.1. A different class of problems for quantum annealing

A computationally hard problem, key for some relevant machine learning

tasks, is the estimation of averages over probabilistic models defined in terms

of a Boltzmann distribution

PB(s) =1

Zexp

∑i,j

Wijsisj +∑i

bisi

, (9)

where Z is the normalization constant or partition function, s = {s1, . . . , sN}denotes a configuration of binary variables, and Wij and bi are the parameters

specifying the probability distribution.

Sampling from generic probabilistic models, such as PB(s) in Eq. (9), is

hard [25] in general. For this reason, algorithms relying heavily on sampling

are expected to remain intractable no matter how large and powerful classical

computing resources become. Even though quantum annealers were designed

for challenging combinatorial optimization problems, it has been recently recog-

nized as a potential candidate to speed up computations that rely on sampling

by exploiting quantum effects, such as quantum tunneling [26, 27].

4.3.2. Quantum-assisted learning of Boltzmann machines

Indeed, some research groups have recently explored the use of quantum

annealing hardware for the learning of Boltzmann machines and deep neural

networks (see [26, 28] and references therein). The standard approach to the

learning of Boltzmann machines relies on the computation of certain averages

that can be estimated by standard sampling techniques, such as MCMC. An-

other possibility is to rely on a physical process, like quantum annealing, that

naturally generates samples from a Boltzmann distribution. In contrast to their

use for optimization, when applying quantum annealing hardware to the learn-

ing of Boltzmann machines, the control parameters (instead of the qubits’ states)

are the relevant variables of the problem. The objective is to find the optimal

21

control parameters that best represent the empirical distribution of a given

dataset.

These ideas are framed within a hybrid quantum-classical computing paradigm.

Given a classical machine learning infrastructure, the idea is to replace the soft-

ware module that generate samples, e.g., via MCMC, with a quantum annealing

process. This quantum sampling module could be similarly employed in other

domains where sampling is useful. Thus, demonstrating quantum speedup for

sampling would have broad implications.

In recent work [26], the QuAIL team has demonstrated how to properly use

a quantum annealer by overcoming critical challenges such as the instances-

dependent temperature estimation. In fact, while the probability distribution

PB(s) in Eq. (9) is specified by parameters Wij and bi, the control parameters

of a quantum annealer are instead Jij = Teff Wij and hi = Teff bi. According

to quantum dynamical arguments [27], Teff is an instance-dependent effective

temperature, different from the physical temperature of the device. Unveiling

this unknown temperature is key to effectively using a quantum annealer for

Boltzmann sampling. By introducing a simple effective temperature estimation

algorithm [26], it was possible to successfully use the D-Wave 2X system for

the learning of a special class of restricted Boltzmann machines that can serve

as a building block for deep learning architectures. Experiments run using a

synthetic dataset showed that the quantum-assisted algorithm outperformed in

terms of quality (i.e., the value of the likelihood reached) the standard classi-

cal algorithm named CD-1 and approached the performance of CD-100, which

takes about 100 times more computational effort than CD-1 (See [26] for details).

Complementary work that appeared roughly simultaneously showed that quan-

tum annealing can be used for supervised learning in classification tasks [28].

These results are encouraging, but there remain numerous challenges before

the full potential of quantum annealing hardware for sampling problems can

be harnessed. While each future generation will no doubt be an improvement,

hardware advances alone will not suffice. The QuAIL team is therefore develop-

ing algorithmic strategies to address these other problems, with promising initial

22

results. For example, we recently experimentally demonstrated [29] the feasibil-

ity of a fully unsupervised machine learning application by successfully training

our quantum annealer, using up to 940 qubits, to generate, reconstruct, and

classify images that closely resemble (low resolution) handwritten digits, among

other synthetic datasets. We showed a Turing test (see Fig. 4 in [29]) to chal-

lenge people to distinguish between handwritten digits and digits generated by

the quantum device; most people we informally showed this Turing test either

failed or found it difficult. To reach this milestone, we implemented densely

connected hardware-embedded models that are more robust to noise and more

efficient to learn with state-of-the-art quantum annealers.

The ultimate question that drives this endeavor is whether there is quantum

speedup in sampling applications. Current experience with the use of quantum

annealers for combinatorial optimization suggest the answer is not straightfor-

ward. This work is part of the emerging field of quantum machine learning [30],

an essentially unexplored territory where quantum annealing might have a large

impact in the near term.

4.4. Best practice programming and compilation techniques

These explorations have spurred QuAIL to design advanced techniques to

guide programming and improve performance. Software calibration methods

devised by the team are described in [31]. In [5], we compare different mappings

and in [32], we present advanced techniques to intelligently select gauges based

on small numbers of trial runs that often improve performance by an order of

magnitude. Compilation strategies for quantum annealers, including guidelines

for optimally setting the strength of JF are discussed in [5, 11, 12]. Furthermore,

we have identified certain common structures in the QUBO representations of

many applications because different constraints often have similar forms [5].

5. Physics of quantum annealing

This section discusses results clarifying the role of various processes in quan-

tum annealing that suggest where to look for potential quantum speedup and

23

where such an advantage would be unlikely. So far, we have been informal about

what we mean by quantum speedup. However, knowing the different types of

quantum speedup is helpful in assessing results related to the computational

power of quantum annealing. It is also necessary to improve our understanding

of potential classes of problems for which such a quantum device can excel.

5.1. Background on quantum annealing

The target of quantum annealing is to optimize a function of QUBO form, as

in Eq. (1). The cost function has a physical realization in a system comprising

quantum bits (qubits) where each binary variable is encoded as a qubit. The

coefficients (ai) translate into bias fields applied on the qubits and (bi,j) is rep-

resented as the coupling strength between two qubits. The cost function thus

corresponds to a Hamiltonian, H1, as in Eq. (2), which describes the energy

of the system. The Hamiltonian bears strong similarity with the cost function.

However, while in the classical cost function the binary variables can take value

either 0 or 1, in a Hamiltonian the qubit is allowed to be (and in a physical

quantum system, can be) in a superposition of these two states α|0〉 + β|1〉.The optimization problem translates into finding the ground state of the Hamil-

tonian, i.e., the eigenstate of the lowest eigenvalue of H1. In order to do so,

quantum annealing introduces quantum fluctuation in the system, represented

as a non-commuting term in the Hamiltonian, H0. A typical H0 easy to prepare

physically is H0 =∑j σ

xj where each σxj swaps states 0 and 1 on the j-th qubit.

The weight of H0 with respect to H1 is the strength of the fluctuation. The

initial state of the system is one with all possible classical configurations that

are equally likely. The system starts with a strong quantum fluctuation that

gradually quenches. The quantum fluctuation provided by H0 allows the dy-

namics to explore a larger region of the search space and gradually concentrate

(with large probability) at the global minimum. At the beginning of the search,

the initial state is very far from the global minimum but a large fluctuation

allows the system a better chance to accept a state that is energetically higher;

thus allowing a more extensive search of the solution space. As the annealing

24

progresses, the fluctuation is tuned down and the system spends more and more

time around the global minimum, eventually staying there once the fluctuation

disappears. This process resembles simulated annealing where the quantum

fluctuation replaces the thermal fluctuations.

Another perspective of the same process is to view the total Hamiltonian as

slow moving and time dependent. If the Hamiltonian is varying slowly enough,

the system will follow its instantaneous eigen-state (this is known as the adi-

abatic theorem). Since the initial state is actually the ground state of H0, a

slow tuning would eventually result in the ground state of the problem Hamil-

tonian, H1. A key question is: how slow is slow enough? During the evolution

when there is another energy level close to the ground state and if the change

of Hamiltonian is not slow enough, there is a risk the system would jump to

the higher level and never return, and the algorithm would fail. The closer the

two energy levels are, the slower the Hamiltonian must vary in order to mitigate

this risk. The spectral gap (the minimal distance between the two energy levels)

plays a crucial role in quantum annealing.

Ref. [33] defines four classes of quantum speedups:

• Provable quantum speedup: It is rigorously proven that no classical algo-

rithm can scale better than a given quantum algorithm.

• Strong quantum speedup: The quantum heuristic is faster than any known

classical algorithm. This type of speedup has been established for dozens

of special-purpose algorithms, with Shor’s polynomial-time algorithm for

factorization being the most prominent. The best classical algorithm may

be continually evolving, as is the case for most areas in which classical

heuristics prevail; the ICAPS (International Conference on Automated

Planning and Scheduling) planning competition and the SAT competition

generally see new algorithms every year.

• Potential quantum speedup: The quantum speedup is in comparison to a

specific classical algorithm or a set of classical algorithms.

25

• Limited quantum speedup: There is a quantum speedup only if the quan-

tum heuristic is compared to the closest classical counterpart.

A finer-grained classification, which takes into account the type of classical

algorithm used in the comparison, has been proposed in [34].

To better understand where quantum annealing may confer an advantage,

it is important to appreciate its major sources of error. The algorithm may fail

to find a solution due to escape from the ground state via either non-adiabatic

transitions or decoherence processes. Yet another possibility is that the ground

state does not correspond to the optimal solution due to control noise. In the

following, we review some of the recent developments in assessing the impact of

these error mechanisms.

5.2. Quantum annealing bottlenecks

Some insight into the relative performance of quantum annealing can be

gained by studying random optimization problems using the tools of the the

statistical mechanics. Absent noise, non-adiabatic transitions can be prevented

only if the annealing proceeds slowly across points where the gap ∆E that sepa-

rates the instantaneous ground state from excited states becomes small (taking

at least time t ∝ h/∆E). The most widely discussed bottleneck, where the

gap reaches a local minimum, is the quantum phase transition. Some of the

computationally hardest problems exhibit a discontinuous (first order) phase

transition, where the gap is exponentially small. In a common scenario, the

ground state wavefunction abruptly changes from being a superposition of a

large number of spin configurations to being nearly localized near a global min-

imum. If the transverse field is lowered too fast, the algorithm performs no

better than a random guess.

Continuous (second order) phase transitions scale better, although strong

fluctuations of disorder (randomness of the parameters of the problem) can still

make the gap scale as a stretched exponential (exponential in some fractional

power of problem size). This still leaves a large swath of problems — most

amenable to quantum annealing — where the disorder is irrelevant at the critical

26

point (phase transition) so that the gap there is only polynomially small. Recent

work [35] addresses this practically relevant scenario and finds that after the

phase transition bottleneck, the algorithm encounters further bottlenecks with

gaps that scale as a stretched exponential.

As annealing progresses, the number of spin configurations with significant

amplitudes decreases until the wavefunction is completely localized. This is

roughly equivalent to having a partial assignment of variables: An increasing

fraction of binary variables have converged to a definite value, while the re-

maining variables are in a superposition state. At times, a state with a different

assignment of already fixed variables becomes more energetically favorable, and

a large number of variable have to flip simultaneously in a multi-qubit tunnel-

ing, which is the source of ”hard” bottlenecks described above. This process is

analogous to ”backtracking” in classical search algorithms.

The major finding is that the number of tunneling bottlenecks is proportional

to the logarithm of problem size. In practice, as the problem size increases, the

time complexity of quantum annealing will exhibit a crossover from polynomial

scaling (when the phase transition bottleneck is dominant) to exponential (when

the expected number of ”hard” bottlenecks exceeds one). This size threshold

is related to the ”density” of spin glass bottlenecks. Similar concept can be

introduced for other heurstic search algorithms, such as simulated annealing.

The bottleneck density can thus be used as a metric of performance indicating

problem sizes above which the time complexity increases exponentially.

Interestingly, the minimum requirement for the annealing time is to avoid

non-adiabatic transition at the phase transition (polynomial scaling). As it turns

out, for fully coherent annealing, having one long annealing cycle versus choos-

ing the best out of repeated short cycles results in identical time-to-solution (as

long as annealing time exceeds the aforementioned minimum). Shorter anneal-

ing times minimize the effects of decoherence and have been favored in most

experimental studies on the D-Wave hardware.

Coupling to the environment affects these results in multiple ways. First, it

changes the universality class of the phase transition, worsening scaling of the

27

minimum annealing cycle [36]. Second, it suppresses multi-qubit tunneling since

in addition to flipping qubits, corresponding environmental degrees of freedom

have to adjust. If quantum-mechanical tunneling is strongly suppressed, equi-

librium may be reached via thermal excitation due to finite temperature. In this

regime, performance would paradoxically improve with increasing temperature

as the system becomes more classical.

5.3. Multi-qubit co-tunneling

Multi-qubit quantum co-tunneling is expected to be a key microscopic mech-

anism responsible for quantum speedup in quantum annealing. In the following,

we consider limited speedup; i.e., speedup compared to simulated annealing.

Realistic hardware is subject to intrinsic noise that affects the quantum dy-

namics of the system, and therefore needs to be considered when evaluating

the efficiency of quantum annealing hardware. The effect of hardware noise is

twofold: (1) Coupling to noise allows inelastic processes, prohibited by energy

conservation in the closed system. Inelastic relaxation provides an efficient route

to a local minimum within a convex region of the potential energy landscape.

(2) Dephasing noise leads to loss of coherence between the states on different

sides of the barrier, resulting in an incoherent tunneling regime, and, in the

strong coupling regime, causes renormalization of the tunneling rate.

In the case of the flux qubits of the D-Wave system, the typical decoherence

time (a measure of how long quantum features of a single qubit can be main-

tained, specifically the characteristic decay time of the off-diagonal elements of

the qubits density matrix) is of the order of nanoseconds to tens of nanoseconds,

which is shorter than the minimum run time of the annealing schedule, 5 mi-

croseconds. Nevertheless, D-wave annealers demonstrate signatures of quantum

many-body dynamics, particularly incoherent multi-qubit quantum tunneling

and evidence of 8-qubit tunneling has been reported [37]. In the course of quan-

tum annealing, the dynamics of the device is limited to low-energy multi-qubit

superposition states, which are more robust against the effects of noise and deco-

herence than single qubit states. In this regime, single qubit excitations caused

28

by noise local to each qubit are strongly suppressed by the strong qubit-qubit

coupling energy. At the same time, slow fluctuations of local magnetic flux re-

sult in a time-dependent spectrum of the multi-qubit low-energy states, which

introduces decoherence of the multi-qubit dynamics.

In the vicinity of the algorithm’s bottlenecks, quantum annealing hardware

realizes incoherent tunneling [37]. Different tunneling regimes are determined

by comparing the quantum tunneling rate near the computational bottleneck

to the characteristic dephasing rate. In a common regime, the tunneling rate

near the bottleneck is exponentially small, while the dephasing rate is at least

of order one. In this regime, quantum tunneling can be only incoherent in

nature [38]: an analog of the decay of a metastable state into a continuous

spectrum encountered in nuclear physics and chemistry, as opposed to a coherent

superposition of states on two sides of a potential barrier. The incoherent regime

is characterized by a quadratic slowdown of quantum tunneling. Nevertheless,

there exist classes of problems where limited polynomial speedup is possible in

this regime, particularly in cases where the shape of the potential barrier favors

quantum tunneling over classical over-the-barrier escape, such as when barriers

are tall and thin [39].

An alternative [40], operational also in the case of thick barriers where the

usual intuition would favor classical escape, is the class of problems character-

ized by exponential degeneracy of the metastable state separated by a barrier

from the global minima. The latter is typical for NP-hard problems; a common

feature of classical mean-field spin glass models [41] is a polynomial number

of the global minima separated by large potential barriers from an exponential

number of metastable states. In such a landscape, simulated annealing slows

down exponentially due to an additional entropic barrier associated with escap-

ing the exponentially degenerate set of metastable states. In contrast, in the

course of quantum annealing, the transverse field splits the degeneracy of the

classical problem and thereby avoids the additional entropic barrier.

To better understand multi-qubit tunneling processes, we developed an in-

stantonic calculus for analytical treatment of the thermally-assisted tunneling

29

decay rate of metastable states in fully-connected quantum spin models [42, 43].

The tunneling decay problem can be mapped onto the Kramers escape problem

of a classical random dynamical field. This dynamical field is simulated effi-

ciently by path integral Quantum Monte Carlo (QMC). We show analytically

that the exponential scaling with the number of spins of the thermally-assisted

quantum tunneling rate and the escape rate of the QMC process are iden-

tical [44]. This analytical result complements prior numerical work [45] and

provides an explanatory model. This effect is due to the existence of a dom-

inant instantonic tunneling path. We solve exactly the nonlinear dynamical

mean-field theory equations for a single-site magnetization vector that describe

this instanton trajectory. We also derive scaling relations for the “spiky” bar-

rier shape when the spin tunneling and QMC rates scale polynomially with the

number of spins while a classical over-the-barrier activation rate scales expo-

nentially.

5.4. Role of noise

Intrinsic noise cannot be eliminated from real quantum devices: manufac-

turing imperfections, as well as thermal fluctuations, induce quantum dephasing

and decoherence (see Section 6). Noise can sometimes be helpful (thermal fluc-

tuations are responsible for the thermally-assisted annealing effects discussed

earlier), but can cause quantum devices to work far from their ideal state, lim-

iting the actual performance and hiding any potential quantum speedup.

In addition, control noise can change the target Hamiltonian H1 with the

consequence that the target solution is no longer in the ground subspace of

H1. In this case, even a perfect quantum device, subject only to control noise,

would find a “false” ground state, which could be far from any target solution.

The maximum noise that can be added to H1 before the target solutions do

not belong to the ground subspace of H1 is called resilience [46, 47]. In general,

resilience can be increased by properly rescaling the energy of H1. Real quantum

devices, however, have a limited range of energies so the resilience cannot be

completely neglected. Recent work shows that a low resilience could hide a

30

quantum speedup [46].

6. Quantum annealing hardware

To date, the most significant progress in quantum annealing hardware is

based on the engineering of quantum superconducting circuits with macroscopic

collective variables (e.g., electric charge and magnetic flux) exhibiting quantum

coherence. Here we review basic design and operational principles of such cir-

cuits, focusing on different types of superconducting qubits, inter-qubit coupling,

and decoherence processes caused by various sources of the environmental noise.

6.1. Quantization of electric circuits with Josephson junctions

Let us briefly describe quantization of zero-resistance superconducting cir-

cuits, which is based on the lumped element method [48, 49, 50]. We can

represent a circuit using two alternative sets of variables: current and voltage

(I(t) and V (t)) or charge and flux (Q(t) and Φ(t)), connected with each other

via the relations I = dQ/dt and V = dΦ/dt. Let us start with the simplest

circuit such as an LC oscillator (see Fig. 6(a)), whose dynamics is governed

by the Kirchhoff’s laws IL = IC ≡ I and VL + VC = 0. Using VL = LdI/dt

and VC = QC/C, one obtains the equation of motion I + ω2LCI = 0, where

ωLC = 1/√LC is the characteristic frequency for classical current (and voltage)

oscillations. The magnetic flux Φ and charge Q are governed by similar equa-

tions, e.g., Φ +ω2LCΦ = 0. Using variables (Q,Φ) one can express the equations

of motion in the Hamiltonian form, Φ = ∂H/∂Q and Q = −∂H/∂Φ, where the

classical Hamiltonian function is H = Q2/2C + Φ2/2L. Following the standard

quantization procedure, we replace classical variables with corresponding oper-

ators, introduce the commutator[Φ, Q

]= ih, and arrive at the Hamiltonian of

a quantum harmonic oscillator, H = Q2/2C + Φ2/2L, describing the quantized

electromagnetic modes of a macroscopic LC circuit with equidistant energy lev-

els, En = hωLC(n+ 1/2) with n = 1, 2 . . .. Clearly, this energy spectrum is not

suitable for an implementation of a two-level qubit.

31

Yw(a) (b) (c)

I

VCVL

Φext Φext

-

Figure 6: (a) Lumped element model for LC oscillator with current I and voltages VC = −VJ .

(b) Tunable SQUID loop biased by external flux Φext. (c) Effective circuit of a qubit.

In order to separate two well-defined levels that can be used as logical states

|0〉 and |1〉, one should employ a non-harmonic circuit with almost negligible

coupling of the qubit levels and the rest of the spectrum. A natural solution

is to introduce a Josephson junction as a nonlinear and non-dissipative element

of the circuit. Josephson junctions are formed by two superconductors weakly

connected through a high barrier. Within the lumped element approach, they

are described by the current-voltage characteristics IJ = I0 sin(2πΦ/Φ0) where

I0 is a critical current and Φ0 = πh/e is the flux quantum. Analysis of different

realizations of a qubit is based on the Kirchhoff’s laws and on the description

of the junction’s contributions in terms of IJ and VJ (or Φ).

A tunable qubit is realized if one replaces a single Josephson junction by a

SQUID loop formed by two parallel junctions biased by an external flux, Φext

(see Fig. 6(b)). The current passing through the SQUID is given by IJ =

I0 cos(2πΦext/Φ0) sin(2πΦ/Φ0) [51], which can be thought of as an effective

junction with tunable critical current Ieff = I0 cos(2πΦext/Φ0) controlled by

Φext. A typical tunable qubit can be represented as an effective junction shunted

by a linear circuit with admittance Yω (see Fig. 6(c)). Below we consider two

basic types of such qubits shunted by either LC oscillator (flux qubit) or a

capacitor (charge qubit).

6.2. Hamiltonians of flux and charge qubits

Effective Josephson junctions, inductance and capacitance, connected in par-

allel, form a flux qubit (see Fig. 7(a)). The circuit is governed by the Kirchhoff’s

32

(b)

IJ

VC IC

VL ILVJ

It

VCVJ

(a)

Vf

2πΦ /Φ0

Figure 7: Effective circuits and potential energies vs. flux for: (a) Flux qubit (tunable Joseph-

son junction shunted by LC oscillator); and (b) Junction shunted by capacitor only (charge

qubit).

laws for currents IJ,L,C and voltages VJ,L,C : IJ = IC + IL, VJ + VC = 0, and

VJ + VL = 0. Using these relations, we obtain the equation of motion for a flux

Φ threading through the device as: CΦ + Φ/L+ Ieff sin(

2πΦΦ0

)= 0, which leads

to the following Hamiltonian of a flux qubit

H =Q2

2C+

(Φ− Φx)2

2L− Φ0Ieff

2πcos

2πΦ

Φ0. (10)

Here we assumed that the inductance loop L can be biased by an additional

external flux Φx applied through inductive coupling. The first (capacitance)

term Q2/2C in Eq. (10) can be interpreted as a kinetic energy while the second

and third terms describe a potential formed by inductance and Josephson terms,

respectively.

For further consideration, it is convenient to introduce dimensionless flux

φ = 2πΦ/Φ0 + π and charge operators q = −id/dφ. Then the Hamiltonian in

Eq. (10) can be expressed as

H = 4EC q2 + EL

(φ− φx)2

2+ Eeff

J cos φ, (11)

and it is different from the LC oscillator by adding the effective energy of Joseph-

son junction, EeffJ = Φ0Ieff/2π. We also introduce here the capacitance and in-

ductance energies, EC = e2/2C and EL = (Φ0/2π)2/L and φx = 2πΦx/Φ0 + π.

The Hamiltonian in Eq. (11) corresponds to a particle with kinetic energy pro-

portional to EC and potential energy determined by the interplay between EL

33

and EeffJ through the ratio β = Eeff

J /EL = 2πIeffL/Φ0. If β < 1, Eq. (11)

describes a single-well anharmonic oscillator, while for β > 1 the double-well

potential emerges and there are two closely-spaced tunnel-split levels defining

a qubit. The quantum dynamics is determined by tunneling between the wells

that can be controlled by variation of the barrier height EeffJ , and by tilting the

two-well potential via the tilt flux φx. Flux qubits described by Eq. (11) are

implemented in D-Wave quantum annealers [51].

A typical charge qubit operates as an open circuit shown in Fig. 7(b). To

derive the Hamiltonian we must omit the inductance term in the equations of

motion, which results in

H = 4EC q2 + Eeff

J cosφ, (12)

and contains only the Josephson (periodic) part of the potential energy. The

eigenvalue problem is reduced to the Mathieu equation. Operational regimes

of various qubits described by the generic Hamiltonian in Eq. (12) drastically

depend on the ratio EJ/EC .

Several types of qubits have been realized during the last two decades. The

simplest charge qubit, comprised of a voltage source in series with a Josephson

junction (the Cooper pair box ), had been implemented in [52]. Because of the

large charging energy, EJ/EC � 1, the two charge states different by a single

Cooper pair are the working states of this qubit. Unfortunately, the Cooper pair

box is highly sensitive to the charge noise. To overcome this difficulty, another

qubit called the transmon was developed [53]. The transmon is derived from

the Cooper pair box, but it operates in a different regime of EeffJ /EC � 1. It

benefits from the fact that its charge dispersion and noise sensitivity decreases

exponentially with EJ/EC . Tuning EeffJ controls the amplitude of the potential,

which forms a periodic array of minima and maxima shown as red and blue

regions of a contour plot in Fig. 7(b). Since EJ/EC � 1, tunneling between

different minima is greatly suppressed and the qubit is realized at an arbitrary

minimum where the lower states are unevenly spaced due to the nonparabolicity

of the cosine potential. Therefore, one can manipulate the lowest pair of levels

34

“passive”—they seek to maintain coherence onlylong enough to entangle quantum bits or demon-strate some rudimentary capability before, inev-itably, decoherence sets in. The next stages ofQIP require one to realize an actual increase inthe coherence time via error correction, first onlyduring an idle “memory” state, but later also inthe midst of a functioning algorithm. This requiresbuilding new systems that are “active,” usingcontinuousmeasurements and real-time feedbackto preserve the quantum information through thestartling process of correcting qubit errors with-out actually learning what the computer is calcu-lating. Given the fragility of quantum information, itis commonly believed that the continual task oferror correctionwill occupy the vastmajority of theeffort and the resources in any large quantumcomputer.

Using the current approaches to error correc-tion, the next stages of development unfortunate-ly demand a substantial increase in complexity,requiring dozens or even thousands of physicalqubits per bit of usable quantum information, andchallenging our currently limited abilities to de-sign, fabricate, and control a complexHamiltonian(second part of Table 1). Furthermore, all of theDiVincenzo engineering margins on each pieceof additional hardware still need to bemaintainedor improved while scaling up. So is advancing tothe next stage just a straightforward engineeringexercise of mass-producing large numbers of ex-

actly the same kinds of circuits and qubits thathave already been demonstrated? And will thismean the end of the scientific innovations thathave so far driven progress forward?

We argue that the answers to both questionswill probably be “No.” The work by the com-munity during the past decade and a half, leadingup to the capabilities summarized in the first partof Table 1, may indeed constitute an existenceproof that building a large-scale quantum com-puter is not physically impossible. However, iden-tifying the best, most efficient, and most robustpath forward in a technology’s development is atask very different frommerely satisfying oneselfthat it should be possible. So far, we have yet tosee a dramatic “Moore’s law” growth in the com-plexity of quantum hardware. What, then, are themain challenges to be overcome?

Simply fabricating a wafer with a large num-ber of elements used today is probably not thehard part. After all, some of the biggest advan-tages of superconducting qubits are that they aremerely circuit elements, which are fabricated inclean rooms, interact with each other via con-nections that are wired up by their designer, andare controlled and measured from the outsidewith electronic signals. The current fabricationrequirements for superconducting qubits are notparticularly daunting, especially in comparison tomodern semiconductor integrated circuits (ICs).A typical qubit or resonant cavity is a few milli-

meters in overall size, with features that aremostly a few micrometers (even the smallestJosephson junction sizes are typically 0.2 mm ona side in a qubit). There is successful experiencewith fabricating and operating superconductingICs with hundreds to thousands of elements ona chip, such as the transition-edge sensors withSQUID (superconducting quantum interferencedevice) readout amplifiers, each containing sev-eral Josephson junctions (52), or microwave ki-netic inductance detectors composed of arrays ofhigh-Q (>106) linear resonators without Josephsonjunctions, which are being developed (53) and usedto great benefit in the astrophysics community.

Nonetheless, designing, building, and operat-ing a superconducting quantum computer presentssubstantial and distinct challenges relative to semi-conductor ICs or the other existing versions ofsuperconducting electronics. Conventionalmicro-processors use overdamped logic, which providesa sort of built-in error correction. They do notrequire high-Q resonances, and clocks or narrow-band filters are in fact off-chip and provided byspecial elements such as quartz crystals. There-fore, small interactions between circuit elementsmay cause heating or offsets but do not lead toactual bit errors or circuit failures. In contrast, anintegrated quantum computer will be essentiallya very large collection of very high-Q, phase-stableoscillators, which need to interact only in the wayswe program. It is no surprise that the leadingquantum information technology has been andtoday remains the trapped ions, which are thebest clocks ever built. In contrast with the ions,however, the artificially made qubits of a super-conducting quantum computer will never be per-fectly identical (see Table 1). Because operationson the qubits need to be controlled accurately toseveral significant digits, the properties of eachpart of the computer would first need to be char-acterized with some precision, have control sig-nals tailored to match, and remain stable whilethe rest of the system is tuned up and then op-erated. The need for high absolute accuracy mighttherefore be circumvented if we can obtain a veryhigh stability of qubit parameters (Table 1); recentresults (43) are encouraging and exceed expecta-tions, but more information is needed. The powerof electronic control circuitry to tailor waveforms,such as composite pulse sequence techniques wellknown from nuclear magnetic resonance (54), canremove first-order sensitivity to variations in qubitparameters or in control signals, at the expense ofsome increase in gate time and a requirement fora concomitant increase in coherence time.

Even if the problem of stability is solved,unwanted interactions or cross-talk between theparts of these complex circuits will still causeproblems. In the future, wemust know and controlthe Hamiltonian to several digits, and for manyqubits. This is beyond the current capability (~1 to10%; see Table 1). Moreover, the number ofmeasurements and the amount of data required

C

L

LJ

e

g E

Φ

Φ

10.5

0.25

0.12

5

0.06

25

100

104

105

103

102

101

LJ /L0

Phase qubit

Hybrid qubit

Flux qubit

FluxoniumQuantronium

Cooper pair box

Transmon

A

f

B

C

0

h2C 2e4LJ

Fig. 2. (A) Superconducting qubits consist of simple circuits that can be described as the parallel com-bination of a Josephson tunnel element (cross) with inductance LJ, a capacitance C, and an inductance L. ThefluxF threads the loop formed by both inductances. (B) Their quantum energy levels can be sharp and long-lived if the circuit is sufficiently decoupled from its environment. The shape of the potential seen by the fluxFand the resulting level structure can be varied by changing the values of the electrical elements. This exampleshows the fluxonium parameters, with an imposed external flux of ¼ flux quantum. Only two of threecorrugations are shown fully. (C) A Mendeleev-like but continuous “table” of artificial atom types: Cooper pairbox (29), flux qubit (33), phase qubit (35), quantronium (37), transmon (39), fluxonium (40), and hybridqubit (41). The horizontal and vertical coordinates correspond to fabrication parameters that determine theinverse of the number of corrugations in the potential and the number of levels per well, respectively.

www.sciencemag.org SCIENCE VOL 339 8 MARCH 2013 1171

SPECIALSECTION

on

Mar

ch 2

5, 2

013

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

EJ /EC

EL /EJ

FIG. 2 (color online). Different types of superconducting qubits. The basic types are the first three ones from above. The bottom three canbe thought of as improved versions, where additional components have been added.

628 Xiang et al.: Hybrid quantum circuits: Superconducting . . .

Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013

(a) (b)

Figure 8: (a) Summary of the basic types of superconducting qubits [54]. (b) Ratios of energies

EL/EJ and EJ/EC for different types of qubits (Mendeleev-like table) [55].

as in the case of a flux qubit. In Fig. 8, we present basic types of qubits [54]

and show typical ratios EL/EJ and EJ/EC for these qubits (“Mendeleev-like

table”) [55]. Selection criteria among various qubits for particular applications

are determined not only by internal device parameters but also by their coupling

properties and tolerance to the environmental noise.

6.3. Inter-qubit coupling

Controllable couplings between qubits is at the heart of any quantum com-

puting application. The simplest and most commonly used couplers are based

on linear superconducting circuits; e.g., mutual inductances or capacitances,

as shown in Figs. 9(a) or 9(b). A typical multi-qubit system is described by

an anisotropic Heisenberg Hamiltonian: H =∑i,αBiασiα +

∑i,j,α(i 6=j)

Jαij σiασjα,

where σiα are pseudo-spin Pauli matrices in a qubit 2×2 Hilbert space, Biα are

the components of local fields, and Jαij are exchange coupling parameters. Mech-

anism of inductive coupling between flux qubits i and j via mutual inductance

Mij = Mji (Fig. 9(a)) is straightforward: if Mij 6= 0, the external flux from

qubit i threads through qubit j loop (or vice versa) and affects the energy lev-

35

this system can be described by the Hamiltonian in Eq. (4),and the strength of the coupling between the charge qubit andthe SC resonator can in principle reach the ultrastrong-coupling regime (Devoret, Girvin, and Schoelkopf, 2007).A similar structure (see Fig. 4) and mechanism are also usedfor the electric coupling of phase qubits with SC resonators(Sillanpaa, Park, and Simmonds, 2007; Hofheinz et al., 2008,2009), where the phase qubits are placed on the two sides ofthe transmission line and coupled to it via capacitors, sittingon two antinodes of the electric field. The photon in the CPWresonator acts as a quantum bus that transfers quantum statesbetween the two phase qubits.

Flux qubits can also couple to CPW resonators via theinduced magnetic field (Yang, Chu, and Han, 2003, 2004;Niemczyk et al., 2010; Peropadre et al., 2010), as shown inFig. 4. A flux qubit placed at or near an antinode of thestanding wave of the current on the SC wire can stronglycouple to the SC resonator via the mutual inductance. In

such a SC circuit, the vacuum Rabi splitting in the trans-mission spectrum was observed, which means that strongcoupling was achieved. Furthermore, by placing an addi-tional Josephson junction at the central SC wire, where theflux qubit is fabricated, the inductive coupling between thequbit and the resonator can be enhanced and can bringthe system to the ultrastrong-coupling regime (Niemczyket al., 2010).

The other type of resonators, LC resonators, can also beintegrated into SC circuits and can couple to charge and phasequbits via capacitors (electric field) or flux qubits via themutual inductance (magnetic field); see Fig. 4. For example,in flux qubits, the lowest two quantum states, which haveclockwise and anticlockwise supercurrents in the qubit loop,are used to denote the two basis states of the qubit. Byemploying the magnetic field produced by the current, theflux qubit can strongly couple to the LC circuit via a largemutual inductance between them. Such flux qubit-resonator

Chargequbit

Fluxqubit

Phasequbit

LC oscillator Coplanar waveguide resonator

(a)

(b)

(c)

Electric field

Magnetic field

FIG. 4 (color online). Schematic diagrams of LC resonators (second column) and coplanar waveguide resonators (third column) coupled tothree types of superconducting qubits.

632 Xiang et al.: Hybrid quantum circuits: Superconducting . . .

Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013

McCc

(a) (d)(c)

(b)

(e)

H = !i=1

2 " Qi2

2Ci− EJi cos#!i$% + !

nUn

#!n − !nx$2

2, #2$

where Ci and EJi= Ici"0 /2# represent the capacitance andJosephson energy of junction i, respectively, and&"0!i /2# ,Qj'= i$%ij. The inductive terms originate from thetwo closed loops with n! (co,act), Lact*L+Lco /4, and Un*#"0 /2#$2 /Ln. The actuator and control loop phases aredefined as !act*#!1+!2$ /2 and !co*!1−!2, respectively.Hamiltonian #2$ can be reduced to an effective one-dimensional system if Lact&Lco because the plasma energyof the control loop will then be much higher than that of theactuator loop. Setting !co=!co

x and combining the Josephsonterms,

H +Qact

2

2Cp+ V#!act$

V#!act$ = Uact, #!act − !actx $2

2− 'eff cos#!act − !act

0 $-'eff *

2#LactIc+

"0cos.!co

x

2/01 + " Ic−

Ic+tan.!co

x

2/%2

!act0 * − arctan" Ic−

Ic+tan

!cox

2% , #3$

where Ic(* Ic1( Ic2 and Cp=C1+C2. Hamiltonian #3$ is ho-mologous to that of an rf-SQUID whose single junction pos-sesses a critical current that is a function of !co

x and whosephase has been shifted by !act

0 .Let the device described by Eq. #3$ be connected to two

qubits via mutual inductances Mco,1 and Mco,2. The mutualinductance between the qubits will be

Meff = Mco,1Mco,2)#1$, #4$

where )#1$*!Iactp /!"act

x represents the first-order #linear$ sus-ceptibility of the coupler8 and the persistent current flowingabout the coupler actuator loop is

Iactp *

'eff

2#Lact/"0sin#!act − !act

0 $ . #5$

If V#!act$ is monostable and the first excited state can beneglected, then one can replace the operator !act by the valuefor which V is a minimum #dV /d!act=0$

!act − !actx + 'eff sin#!act − !act

0 $ = 0, #6$

which can be solved for !act given arbitrary !actx , thus yield-

ing Iactp #"act

x ,"cox $. Differentiating Eqs. #5$ and #6$ with re-

spect to "actx then yields )#1$

)#1$ *!Iact

p

!"actx =

1Lact

'eff cos#!act − !act0 $

1 + 'eff cos#!act − !act0 $

. #7$

Equation #7$ is similar to Eq. #10$ of Ref. 8, albeit 'eff is afunction of !co

x and junction asymmetry results in a!co

x -dependent phase shift in the cosine terms.While rf-SQUID and CJJ rf-SQUID couplers possess

similar expressions for )#1$, the latter holds two advantages:first, the CJJ coupler can be operated with "act

x =0 and tunedvia "co

x . If Ic− / Ic+*1, then !act0 *1 and Eq. #6$ yields !act

+0. Equation #5$ then predicts that Iactp +0. Thus the CJJ

coupler need not invoke large persistent currents #on the or-der of Ic+$ when being tuned. Second, the CJJ coupler isusable over the range of "co

x for which −min&1,'eff#0$'+'eff#!co

x $,'eff#0$ when "actx =0, where the lower bound

has been imposed by the condition that V#!act$ bemonostable. Thus the utility of the CJJ coupler is not com-promised if 'eff#0$-1. As such, this device is robust againstfabrication variations.

To test the CJJ rf-SQUID coupler, we fabricated a circuitcontaining 8 CJJ rf-SQUID flux qubits,12,13 each inductivelycoupled to its own hysteric dc-SQUID readout,15 and con-nected by a network of 16 CJJ rf-SQUID couplers. The chipwas fabricated from an oxidized Si wafer withNb /Al /Al2O3 /Nb trilayer junctions, four Nb wiring layerscapped with SiN and separated by planarized plasma-enhanced chemical-vapor deposition SiO2. The chip wasmounted to the mixing chamber of a dilution refrigeratorregulated at T=40 mK inside a Sn superconducting mag-netic shield with a residual field in the vicinity of the chip+1 nT. External current biases were provided by room-temperature current DACs whose outputs were low pass fil-tered with fc+5 MHz using a combination of lumped ele-ment and copper powder filters secured to the mixingchamber.

A schematic of a coupler and a pair of qubits is depictedin Fig. 2. The coupler is controlled via bias currents that arecoupled to the device through mutual inductances Mco andMact, respectively. These give rise to the fluxes "co

x and "actx .

The qubits are controlled via fluxes "cjj.x and "q.

x #.=1,2$ asdescribed in Ref. 13. The qubits interact with the coupler viamutual inductances Mco,.. For brevity, we present resultsfrom a single coupler in this Brief Report and note thatMeff#"co

x $ was identical to +5% for all 16 couplers on thischip. For the particular coupler described herein, the relevantqubit critical currents were Iq.

c =3.25(0.01 /A and qubitinductances were Lq1#2$=290#308$(5 pH when "co

x =0.The flux wave forms used to obtain Mact are depicted in

Fig. 3#a$. In this case, "cox was held constant while the de-

tector qubit #.=d$ was annealed in the presence of a pulseon "act

x #t$ of amplitude "acti and a pulse on "q.

x #t$ of ampli-tude "q

d. The sequence involves #i$ initializing the qubit in amonostable potential with no net flux biases, #ii$ setting "act

x

and "q.x , #iii$ raising the detector qubit’s tunnel barrier to

Qubit 1

Mco

Qubit 2Coupler

FactFactx

Mact

Mco,1 Mco,2Fcox

Fq1Fq1x Fq2Fq2

x

Fcjj1x

Fcjj2x

FIG. 2. #Color online$ Schematic of a CJJ rf-SQUID couplerinteracting with two CJJ rf-SQUID qubits.

BRIEF REPORTS PHYSICAL REVIEW B 80, 052506 #2009$

052506-2

Figure 9: Effective circuits for different regimes of interqubit coupling: (a) between flux qubits

via mutual inductance Mc ≡ M12, (b) through inductive loop controlled by SQUID, [56] (c)

between transmons coupled via capacitance Cc, and (d) tunable coupling between transmons

controlled by Josephson junction with nonlinear inductance Lc. [57] (e) Schematic of the

coplanar waveguide resonator (light blue), the transmon qubits and the first harmonic of the

standing wave electric field shown in red. [58, 54]

els. Thus, the longitudinal coupling (proportional to σ1zσ2z) can be expressed

as Jzij ∼ MijIiIj . The direct inductive coupling is not tunable; however, a

tunable coupling strength can be realized if the inductance loop is driven by a

SQUID. An example of such coupling, utilized in D-Wave quantum annealers,

is schematically shown Fig. 9(b) [56]. It is based on bias currents that produce

controlled flux biases.

A circuit diagram of two capacitively-coupled transmons is shown in Fig. 9(c),

and can be analyzed using the lumped element method as above. As a result, the

interaction Hamiltonian for a pair of transmons can be expressed as qiqjC/Cc.

Calculating matrix elements of qi,j within the two-level approximation, we ob-

tain the transverse coupling (proportional to σixσjx) with the coupling param-

eter Jxij ∼√

∆Ei∆Ej(C/Cc), where ∆Ei is level splitting of i-th transmon.

The purely capacitive coupling is not tunable, but the coupling strength can

be controlled using a non-linear coupler with Josephson junction (a tunable in-

ductor). This circuit is depicted in Fig. 9(d), where arrows indicate the flow

of current for an excitation in the left qubit [57]. It is important that the cou-

36

pling be tunable with nanosecond resolution, making this circuit suitable for

various applications ranging from quantum logic gates to quantum simulations.

Similar circuits are employed for readout of a flux qubit state in D-Wave quan-

tum annealers, where each qubit is connected inductively with a quantum flux

parametron (rf -SQUID with a small inductance, a large capacitance and a very

large critical current) [51, 59]. Another approach is to couple all qubits to a

shared passive element (quantum bus) such as a cavity or a coplanar waveguide

resonator (CPW) [54].

6.4. Qubit relaxation and decoherence

Superconducting qubits are macroscopic quantum objects whose generic

quantum properties, such as superposition of states and entanglement, inher-

ently suffer from detrimental effects caused by a macroscopic, noisy environ-

ment [60]. To describe environmental noise phenomenologically, one should

take into account random charge, flux, and Josephson junction noise sources

that modulate lumped elements of the equivalent circuit in the qubit Hamilto-

nians in Eqs. (11) or (12).

After tracing over the environmental variables, the qubit dynamics is gov-

erned by the Bloch equation with two transition rates Γ1 and Γ2 (or times

T1 and T2) describing qubit relaxation and decoherence, respectively. The two

rates are related: Γ2 = Γ1/2 + Γd, where Γd describes dephasing due to the low

frequency noise. The flux qubits (e.g., D-Wave qubits) studied to date suffer

from a low-frequency flux noise due to environmental spins [61, 62]. This leads

to a substantial dephasing rate Γd and, in turn, to a large difference between

the relaxation and decoherence rates, Γ2 ∼ Γd � Γ1. In transmon qubits, the

flux noise is absent and the low-frequency charge noise is suppressed; i.e., the

decoherence rate is low and Γ2 and Γ1 are close to each other.

A particular choice of a qubit depends on its suitability for a given appli-

cation. For instance, quantum annealing requires strong coupling between the

qubits. Therefore, in this case the flux qubit is a preferred choice because a

typical value of the coupling parameter for D-Wave flux qubits is several GHz.

37

On the other hand, the coupling between transmon qubits is much weaker (on

the order of 10 MHz). Thus, the coupling and connectivity requirements of the

quantum annealing outweigh the disadvantages caused by the higher decoher-

ence rate of the flux qubits.

7. Conclusions

The emergence of quantum annealers in the past few years has enabled the

explorations described in this paper. The next few years promise to be yet more

exciting as more sophisticated quantum annealers become available and one sees

the advent of the first universal quantum computers able to run other quantum

heuristic algorithms. The NASA QuAIL team is excited to be at the forefront

of these developments, and looks forward to working with quantum hardware

and algorithms teams from around the world to explore quantum heuristics and

thereby broaden the areas in which quantum computation has clear applications.

8. Acknowledgements

The authors would like to acknowledge support from the NASA Advanced

Exploration Systems (AES) program and NASA Ames Research Center. This

work was supported in part by the AFRL Information Directorate under grant

F4HBKC4162G001, the Office of the Director of National Intelligence (ODNI),

and the Intelligence Advanced Research Projects Activity (IARPA), via IAA

145483. The views and conclusions contained herein are those of the authors

and should not be interpreted as necessarily representing the official policies or

endorsements, either expressed or implied, of ODNI, IARPA, AFRL, or the U.S.

Government. The U.S. Government is authorized to reproduce and distribute

reprints for Governmental purpose notwithstanding any copyright annotation

thereon.

38

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